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Math Help - Need help putting proof into proper form

  1. #1
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    Need help putting proof into proper form

    Show that the relations (A \cup C) \subset (A \cup B) and  (A \cap C) \subset (A \cap B) , when considered together, imply  C \subset B .

    Here is my proof:
    If x \in (A \cup C) then x \in (A \cup B).
    If x \in (A \cup C) then x \in A or  x \in C.
    We want to know about the elements in C so we consider x \in C.
    If x \in C then x \in (A \cup B).
    This means that if x \in C then x \in A, x \in B, or x \in A and x \in B.
    There are three cases.
    First: x \in C, x \in A \implies x \in (A \cap C). From the second relation, x \in (A \cap B), and therefore x \in B.
    Second: x \in C, x \in B \implies x \in B.
    Third: x \in C, x \in (A \cap B) \implies x \in B.
    Thus x \in C \implies x \in B, therefore C \subset B.

    Is this proof complete, correct, clean, nasty?
    Any help is appreciated, especially cleaning it up and making it look all nice with proper notation. Again, if it's wrong it would be more helpful if you could point out where I've went astray.
    Thanks,
    ultros
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  2. #2
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    Disproving Converse

    Also, I'm having difficulty finding an example to disprove the converse. Thanks again.
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  3. #3
    Lord of certain Rings
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    Quote Originally Posted by Ultros88 View Post
    Also, I'm having difficulty finding an example to disprove the converse. Thanks again.
    I think the converse is true:

    That is:

    If C \subset B, A \cup C \subset A \cup B and A \cap C \subset A \cap B

    I think its easy to prove:

    x \in A \cup C \Rightarrow x \in A \text{ or } x \in  C \subset B \Rightarrow x \in A \text{ or } x \in B \Rightarrow x \in A \cup B

    Similarly the other part.


    P.S: What inspired the name Ultros(FF?)?
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  4. #4
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    Converse

    Yeah, I think the converse is true as well, which is funny because the book I'm using seems to want me to prove it false.

    And the name did come from FF by the way. My favourite octopus.
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